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Full text of "Mathematical And Physical Papers - Iii"

ON  THE MOTION  OF  PENDULUMS.                            59
have seen, however, that the results applicable to uniform motion may be deduced as limiting cases of those which relate to oscillatory motion, and consequently, we may make use of the expression for F already worked out.    Writing V for ce^~lnt in the first equation of Art. 20, expressing m in terms of n, and then making n vanish, we get
-F^eirp'paV. .................... (126),
and — F is the resistance required.
This equation may be employed to determine the terminal velocity of a sphere ascending or descending in a fluid, provided the motion be so slow that the square of the velocity may be neglected. It has been shewn experimentally by Coulomb *, that in the case of very slow motions, the resistance of a fluid depends partly on the square and partly on the first power of the velocity. The formula (126) determines, in the particular case of a sphere, that part of the whole resistance which depends on the first power of the velocity, even though the part which depends on the square of the velocity be not wholly insensible.
It is particularly to be remarked, that according to the formula (126), the resistance varies not as the surface but as the radius of the sphere, and consequently the quotient of the resistance divided by the mass increases in a higher ratio, as the radius diminishes, than if the resistance varied as the surface. Accordingly, fine powders remain nearly suspended in a fluid of widely different specific gravity.
and in the present Section ^ has the form SI> sin2 0, where <fr is independent of 0. Hence we get from (6), by integrating partially with respect to 0,
f-
dr \dr-    r2    /* dt
It is unnecessary to add an arbitrary function of r, because if X (r) be such a function which we suppose added to the right-hand member of (c), we must determine X by substituting in (a). The resulting expression for X' (?•) cannot con-tain 6, inasmuch as the expression for dp is an exact differential, but it is composed of terms which all involve cos0 as a factor, and therefore we know, without working out, that these terms must destroy one another. Hence X (r) must be constant, or at most be a function of *, which we may suppose included in n. X (r) will in fact be equal to zero if n be the equilibrium pressure at the depth at which fgdz' vanishes.
* M&rtoires de VInstitut, Tom» in. p. 246,